Damage amplification during repetitive seismic waves in mechanically loaded rocks

Cycles of stress build-up and release are inherent to tectonically active planets. Such stress oscillations impart strain and damage, prompting mechanically loaded rocks and materials to fail. Here, we investigate, under uniaxial conditions, damage accumulation and weakening caused by time-dependent creep (at 60, 65, and 70% of the rocks’ expected failure stress) and repeating stress oscillations (of ± 2.5, 5.0 or 7.5% of the creep load), simulating earthquakes at a shaking frequency of ~ 1.3 Hz in volcanic rocks. The results show that stress oscillations impart more damage than constant loads, occasionally prompting sample failure. The magnitudes of the creep stresses and stress oscillations correlate with the mechanical responses of our porphyritic andesites, implicating progressive microcracking as the cause of permanent inelastic strain. Microstructural investigation reveals longer fractures and higher fracture density in the post-experimental rock. We deconvolve the inelastic strain signal caused by creep deformation to quantify the amount of damage imparted by each individual oscillation event, showing that the magnitude of strain is generally largest with the first few oscillations; in instances where pre-existing damage and/or the oscillations’ amplitude favour the coalescence of micro-cracks towards system scale failure, the strain signal recorded shows a sharp increase as the number of oscillations increases, regardless of the creep condition. We conclude that repetitive stress oscillations during earthquakes can amplify the amount of damage in otherwise mechanically loaded materials, thus accentuating their weakening, a process that may affect natural or engineered structures. We specifically discuss volcanic scenarios without wholesale failure, where stress oscillations may generate damage, which could, for example, alter pore fluid pathways, modify stress distribution and affect future vulnerability to rupture and associated hazards.

Porosity determination. Because porosity and pore structure have a first-order control on material strength [50][51][52] , we chose a texturally homogeneous andesite, and determined the connected porosity of each prepared specimen using an AccuPyc 1340 helium pycnometer with a 100-cm 3 chamber 53 . The pycnometer provided a measure of the skeletal volume (V s ) of each sample (i.e., the volume of solid rock, including isolated pore space, which is scarce in crystalline volcanic rocks) with an accuracy of ± 0.1% of the measured volume. The volume of each core sample (V c ) was then geometrically calculated, and the connected porosity (φ) was determined via: φ = (V c − V s )/V c . The 27 specimens measured had a connected porosity ranging between 11.75% and 13.22% with an average of 12.72% (see Supplementary Table 1).
Mechanical tests. In this study, three types of mechanical experiments were undertaken using an Instron 8800 uniaxial press: (i) uniaxial compressive strength (UCS) tests, (ii) creep tests, and (iii) oscillation tests under creep loads. During the tests, the stress was monitored with an accuracy of 0.02 MPa (estimated from standard error and taking into account load cell accuracy and sample dimensions), and the axial displacement was monitored at an accuracy of 10 -5 mm. Using the monitored displacement at a given load, the axial strain was calculated, after correcting for the machine compliance (of the piston assembly) recorded at the same loading conditions.
For the UCS tests, samples were axially loaded at a constant strain rate of 10 -5 s −1 until they underwent system-scale failure to define the strength of the materials and set the conditions for the subsequent tests. For the creep tests, samples were first loaded at a constant strain rate of 10 -5 s −1 to the desired creep condition (i.e., to stresses corresponding to 60, 65, or 70% of the UCS value of the rock; see dotted lines in Fig. 2). The stress showing that micro-fractures extend from the pores, sub-parallel to the applied principal stress (σ 1 , indicated in (b)) and traverse the groundmass and phenocrysts as a result of the experiment. (f) Fracture length distribution for the intact (light grey) and deformed samples (dark grey) showing that the deformed material hosts a slightly higher number of fractures and thus fracture density. Fractures in the deformed material also attain greater lengths than the pre-deformed material. www.nature.com/scientificreports/ was then held constant for 6 h before unloading at a constant strain rate of 10 -5 s −1 . The oscillation tests were run at the same loading/unloading conditions as the creep tests; however, during the 6-h creep stage, 15 repeated mechanical oscillations of set amplitude were introduced every 22 min. In order to simulate the variable oscillations in strain encountered by volcanic rocks during natural earthquakes, we used a nominal waveform produced by stacking 181 seismic waves from an earthquake swarm (during which both source location and mechanism remain relatively constant) recorded between 01/10/2015 and 15/11/2015 at Unzen volcano, Japan 54 . The stacked waveform was then normalised in both time and absolute amplitude before being integrated into the Wavematrix 2.0 software of Instron. The normalised waveform was then used to control the actuator (i.e., the piston motion) by multiplying the amplitude by the desired peak load, set as a fraction of the load hold condition (2.5, 5.0, and 7.5%; see insert in Fig. 2) around a constant mean load value corresponding to a given creep condition (60, 65, or 70% of the UCS; Fig. 2). The duration of the applied load oscillation matched that of the original stacked waveform, equating to a dominant shaking frequency of ~ 1.3 Hz. We note that our experimental setup can only recreate compression/decompression around that mean load value, and thus we simulate a transverse motion in the direction of the pistons' axis.
Young's modulus determination. As a proxy to assess the resultant changes in mechanical behaviour imparted by creep and mechanical oscillations, the Young's modulus was determined during the loading and unloading phases of each experiment. Similar to the short-term, long-term average method 55 , a bespoke MatLab code was used to fit linear regressions to the stress-strain data comprised in a long window (here 1/4 of the loading/ unloading phase) rolling through the data as well as for 5 smaller windows inside the long window (i.e., 1/20 of the loading/unloading phase). For any given long window, the mismatch in slope with each small window was calculated alongside the coefficient of determination, r 2 . The long window providing the highest r 2 and smallest average mismatch was then used to automatically estimate the Young's modulus. The same code was used for the determination of Young's modulus during the UCS tests.

Acoustic emissions.
To resolve the development of cracking events throughout all mechanical tests, acoustic emissions (AEs) were monitored at rate of 1 GHz using a high sensitivity R50S piezoelectric transducer connected to the curved, long edge of the sample. The AEs were fed to a 2/4/6 preamplifier set to a 20-dB gain, before reaching a PCI-2 data acquisition system developed by Physical Acoustics Corporation (PAC).

P-wave velocity.
To assess the accumulation of physical damage produced by the tests, we measured P-wave velocity (Vp) for all samples preceding and immediately following each test. Two sensors were attached to opposing ends of the cylindrical specimens. The PAC PCI-2 system emitted 100 pulses through a R50S piezoelectric sensor connected to a 2/4/6 preamplifier, set to a gain of 40 dB, and another R50S piezoelectric sensor connected to the PAC PCI-2 system received the pulses. Pulses had 1 μs duration, and were sent every 5 s to avoid interactions between pulses. Vp estimates were then made by selecting the 20 most energetic pulses and finding the time delays between the 2 sensors, which were divided by the length of the sample. For each oscillation test, one amplitude variation is selected, corresponding to ± 2.5% (black), ± 5% (dark grey) or ± 7.5% (light grey) of the original creep load applied. SEM imaging and fracture tracing. To image and quantify the microstructure of the rocks, backscattered electron (BSE) images were taken using a Zeiss GeminiSEM 450 scanning electron microscope (SEM) from the SEM Shared Research Facility at the University of Liverpool. Images with a resolution of 150 dpi were acquired using an accelerating voltage of 10 kV, beam current of 5 nA, and a working distance of 9.9 mm. For each BSE image acquired, the fractures were manually identified and traced using ImageJ. To reduce processing time, each BSE map was cut in 9 different sections of 4.99 × 3.32 mm and tracing was performed at a zoom of 150× using the segmented line tool. Fracture width and average angle were also measured using the same images, albeit at a zoom of 1200×, by averaging the length/angle of three straight lines perpendicular to the length trace. The magnitude of the stress during oscillation was set as a fraction (± 2.5, 5.0 or 7.5%) of the aforementioned nominal target stresses (see black line and insert in Fig. 2). Here, all but one samples remained coherent after these tests; one of the two samples tested at the most extreme conditions (a creep stress of 70% of the UCS and oscillations of 7.5% of the nominal target stress) underwent rupture between the 14th and 15th oscillations.

Results and micro-mechanical interpretations
During creep tests and oscillating stress tests, AEs were monitored as a proxy for the occurrence of microfracturing events 41,56-58 (Fig. 3, Supplementary Fig. 2). In the creep tests, most AEs were generated during the loading phase, and few AEs were detected during the constant load phase (reducing from the onset of the hold; Fig. 3a-c). This indicates that during the initial loading phase, pre-existing micro-fractures shut perpendicular to the primary applied stress direction, σ 1 and micro-fractures were created/ modified parallel to σ 1 , and that during creep very few micro-cracks nucleated and grew at a detectable scale. This is consistent with previous creep experiments that showed that only small amounts of subcritical crack growth occur before reaching tertiary creep deformation 19 . During oscillation tests we noted a similar preponderance for AEs during the loading phase ( Fig. 3d-l). However, we also monitored bursts in AE activity when mechanical oscillations shook the samples (outlined by dotted lines on Fig. 3d-l); we observed that the number of AEs in these bursts increases with both the stress level of background creep and with the amplitude of the oscillations. These bursts in AEs indicate that either new fractures are nucleated and/ or existing cracks propagate and coalesce. For the sample that underwent rupture between the 14th and 15th oscillations, we observed an acceleration in AE before rupture (see Supplementary Fig. 2l). We also observed that the ± 2.5% amplitude tests did not seem to generate AE bursts. Additionally, only the first few oscillation events during the ± 5% amplitude tests registered AE bursts. In both cases, this could be due to the fact that the sensors were not able to record the smallest cracking events and thus we cannot rule out the occurrence of cracking due to these small oscillations.
The creep and oscillation tests result in the observation of inelastic (permanent) strain ( ε i ) in the materials. The amount of inelastic strain during the 6-h creep period of both types of extended tests (creep and oscillation) is determined by the difference between the maximum accumulated strain and the strain at the start of the creep period (i.e., following the initial loading phase; Fig. 4a; see also Supplementary Figs. 3-5 for the raw mechanical data). It is worth noting that the stressing conditions for the creep and oscillation tests largely overlap, as oscillation amplitudes are a small percentage of the creep condition, and thus there are substantial overlaps within the observed mechanical results. However, careful examination of each monitored and measured parameter enables us to establish various trends, which we discuss holistically here. For creep tests, the results show that the amount of inelastic strain increases with applied creep stress (see squares in Fig. 4b; darker colours depicting higher creep loads). When mechanical oscillations were introduced, the sample underwent up to four times more inelastic strain than under creep conditions alone (at least for samples that did not lead to system-scale failure), and more than ten times after the 14th oscillation in the sample that ultimately underwent failure (see stars in Fig. 4b). However, there did not seem to be a clear correlation between the accumulated strain and the amplitude of the oscillations for a given nominal creep stress. Additionally, and contrary to the creep tests, the intensity of the applied creep stress during oscillation tests does not show a correlation with the maximum inelastic strain.
To investigate the development of inelastic strain imparted by individual mechanical oscillations further, we opted to filter out the creep deformation from the oscillation test samples by computing the average strain response at each creep condition (see Supplementary Information: Dynamic creep removal and Supplementary  Fig. 6). We thus first cropped the strain response to each stressing event by finding the most prominent peak in the stress signal; cropping is done by selecting strain data 0.2 s before (100 data points at a sampling rate of 50 Hz) and 30 s after (1500 data points at a sampling rate of 50 Hz) the most prominent stress peak. By filtering out the creep deformation we obtained the zeroed strain response during each oscillation (Fig. 5a)   www.nature.com/scientificreports/ that the zeroed strain signal of each oscillation event became slightly compressed along both amplitude and time axes when compared to the "pristine" first straining event. Using dynamic time warping 59,60 to match each individual oscillation event to the first reference or "pristine" event then allowed us to calculate the inelastic strain accumulated by the sample when the oscillation events occurred (i.e., the length of the shortest warping path between the two strain signals) at the different conditions tested. Examining the strain accumulated by the samples during oscillations, we observe that the first few oscillations (up to 4-6) always accumulate the most strain, regardless of the amplitude or creep stress (Fig. 5b-d). Subsequent oscillations then appear to impart a stable amount of inelastic strain, but in some cases, events progressively accumulate smaller amounts of damage than earlier oscillation events. For the sample that failed (creep stress of 70% of UCS and oscillations of ± 7.5% amplitude), the strain accumulated at each event began to increase again after the initial reduction. It is worth noting that for the sample that underwent failure we considered the last cycle to be the 14th oscillation. Similarly, for a sample tested at a creep load of 60% of UCS and subjected to oscillations of ± 7.5% amplitude (Fig. 5b), each oscillation event imparted more inelastic strain than the previous event, thus showing an acceleration in inelastic strain build up as samples approached failure under high loads, similar to that experienced during tertiary creep 19,61,62 . Although the reason for this specific sample exhibiting this response is unknown, it may be due to a higher starting micro-fracture density that would have promoted fracture coalescence. Finally, the amplitude of the oscillations seems to slightly correlate with the total amount of inelastic strain experienced by a rock (Fig. 5e), as samples subjected to the larger amplitudes tended to record higher total accumulated strains after the last straining event, especially the samples that experienced the largest, ± 7.5% amplitude, oscillations.
To assess the mechanical changes imparted during the different tests, the Young's modulus was calculated during the loading (empty squares and stars) and unloading (filled squares and stars) phases of creep and oscillation tests (Fig. 6a) Table 1). This apparent stiffening may be explained by the closure of microfractures perpendicular to the applied stress during loading, which accommodates more strain at a given stress than the elastic deformation of the rock, and by the nucleation, growth, and coalescence of micro-cracks parallel to the applied stress. Subsequently, as the material is unloaded, asymmetric reverse sliding along these micro-cracks can alter the bulk mechanical response of the material and confer an hysteresis during cycles of loading/unloading in uniaxial conditions 63,64 . Similarly to the results presented above, we observe that oscillation tests show a more drastic change in Young's Modulus, thus indicating that those stressing events contribute alongside creep to introduce damage in the samples.

Induced damage impact on physico-mechanical properties.
Examination of the micro-structure of the starting material and a deformed sample (65%; ± 5%) show that the experiment modified the architecture of the fracture network (Fig. 1b,d-f; see also Supplementary Figs. 7, 8). The original material contains microfractures that are generally restricted to either the groundmass or the phenocrysts (i.e., the fractures do not cross the boundaries; Fig. 1a,c). Additionally, the undeformed sample exhibits a fracture density of 5.3 × 10 6 m −2 , and the longest fracture measured reaches 867 µm (Fig. 1f). In contrast, fractures in the sample that has been subjected to oscillations tend to emanate from heterogeneities and cross both groundmass and phenocrysts alike (Fig. 1d,e). This sample exhibits a fracture density of 7.4 × 10 6 m −2 , and the longest fracture reaches 1861 µm (Fig. 1f). Overall, fractures cross groundmass and phenocrysts without deflection, indicating physical damage a) b)  www.nature.com/scientificreports/ imparted by both creep and fatigue mechanisms in contrast to the pre-existing fractures commonly observed in rocks from Volcán de Colima, which are likely associated with cooling of the lava after eruption 31 . Ultrasonic velocities of the compressional waves, Vp, were used to assess the material's integrity after the tests. The range of Vp was slightly higher following creep tests (Fig. 6b) The increase in spread of Vp following the oscillation tests was more significant than in creep tests, suggesting greater microstructural rearrangement of these samples. Vp was measured along the long axis of the samples, parallel to the principal applied stress, which is also the predominant orientation of induced fractures, therefore their impact on the measured Vp is limited. However, in combination with changes in Young's modulus, the Vp results suggests (Fig. 6) that the passage of mechanical oscillations in a rock mass amplifies the effect of creep deformation and imparts further damage that modifies the microstructure, and importantly increases the anisotropy of the fracture network and the resultant mechanical properties of rocks. The presence of micro-fractures is an important control on permeability 66 , which, in turn, accommodates the presence and flow of pore fluids that affect the stress and, potentially, the resultant behaviour of materials 12 . As the damage created during the extended tests was generally microscopic (i.e., only samples which underwent rupture showed macroscopic damage), we did not anticipate large changes in absolute permeability values 67,68 ; hence, we focused our attention on the pressure dependence of the permeability in the original materials versus experimental products, as fracture geometry (which controls permeability; e.g., aperture, connectivity, tortuosity) is pressure dependent 69 . We quantify the permeability change rate, ⍺, as a metric for relative changes in the fracture characteristics of our samples, by fitting a linear regression through the permeability values as a function of confining pressure (Fig. 7a). The permeability is normalised to the permeability at the minimum confinement of 0.7 MPa to remove the effect of sample variability and allow comparison between samples that were deformed at different conditions (see Fig. 7b). The starting materials exhibit a range of ⍺ due to material variability, with a) b) Figure 6. Changes in Young's modulus and P-wave velocity (Vp) before and after the creep period. (a) Young's modulus during the loading phase (hollow symbols) and unloading phase of creep (squares) and oscillation (stars) tests. (b) Ultrasonic velocities, Vp, estimated in pre-test samples (hollow symbols) and post-test samples deformed in creep (squares) and oscillation (stars) tests. For both creep and oscillation tests, the different shades of blue correspond to different stresses at each creep condition (the darker the shade, the higher the stress). For oscillation tests, the size of the star depicts the amplitude of the stress oscillation, with small stars corresponding to ± 2.5%, intermediate to ± 5%, and large to ± 7.5% of the creep stress set-point. www.nature.com/scientificreports/ values generally above − 0.15 MPa −1 . Following creep tests, we note a clear reduction in ⍺, indicating a higher susceptibility of permeability to reduction under confinement. Following the oscillation tests, ⍺ drops even more substantially, showing a greater increase in the dependence of permeability on confining pressure. The changes in fracture morphology (Fig. 1, Supplementary Figs. 6, 7) explain the observation of increased susceptibility of permeability to confining pressure, as the longer fractures generated parallel to σ 1 (hence perpendicular to confinement in the permeameter) would be more effectively closed under confining pressure. Moreover, there appears to be a positive correlation between ⍺ and the oscillation amplitude. This supports the observations from the monitored AE that mechanical oscillations enhance the effect of creep and impart fracture damage that changes the physico-mechanical properties of rocks, and further indicates that the amplitude of stress oscillations effects the architecture of the permeable porous network.
Impact of sample variability. The failure mechanism of rocks is primarily affected by microstructural arrangements (pore/crystal/micro-fracture size, orientation, geometry, and spatial relationship to one another) 17,50,62,63 . As such, every sample has a unique microstructural arrangement with an inherent variability. Therefore, sample variability may hinder direct sample to sample comparison, limiting our ability to interpret specific contrasts between individual test results. However, when our data are considered as a whole, we observe quantifiable signals and correlated trends that indicate that sample variability only has a minor effect on the results we present here, leading to overlapping ranges of values but not obscuring trends across the datasets. The permeability pressure dependence, or permeability change rate, α, corresponds to the slope of the linear regression. (b) Permeability change rates, α, (as explained in a) calculated before (hollow symbols) and after deformation for samples in creep (squares) and oscillation (stars) tests. For both creep and oscillation tests, the different shades of blue correspond to different stresses at each creep condition (the darker the shade, the higher the stress). For oscillation tests, the size of the star depicts the amplitude of the stress oscillation, with small stars corresponding to ± 2.5%, intermediate to ± 5%, and large to ± 7.5% of the creep stress set-point.

Implications
High-amplitude mechanical oscillations can trigger structural failure in both natural and man-made materials 70,71 . This effect is particularly noteworthy for materials located in the direct vicinity of large stress disturbances (e.g., close to the focus of an earthquake), and it lessens with distance, as attenuation and dispersion reduce the amplitude of the stress oscillation 1 . Our experimental campaign has shown that mechanical oscillations can systematically modify the physical and mechanical properties of rocks, and in extreme cases, prompt rupture. Here, using dynamic time warping on the recorded strain signals, we isolated the effect of fatigue mechanisms caused by individual oscillations from the effect of creep to show the build-up of inelastic strain caused by mechanical oscillations (even when the maximum stress is below the compressive strength of the material). Such mechanical pulses contribute to the generation of micro-cracks sub-parallel to σ 1 and thus act to amplify the effect of time-dependent deformation in materials otherwise subjected to constant creep stresses (tectonic, gravitational). Importantly, although we were able to distinguish between, and quantify, the physical impact of mechanical oscillations during fatigue stressing over that of creep deformation, we could not resolve which specific mechanisms (e.g., wear along crack edges 13,64 , grain indentation 41,42 , etc.) were operating during our experimental campaign. We surmise that the amplification of physical damage by mechanical oscillations can modify the stress fields in rock masses 72 at a wide range of temporal and spatial scales without necessarily prompting rupture or substantial deformation. For example, stress transfer along faults after an earthquake can be attributed to direct fault-fault interactions 73 as well as to the redistribution of fluids and pore pressure 74 that, as we show here, would be facilitated by the creation of damage due to mechanical oscillations. Our results indicate a fracture-driven anisotropy of the samples caused by deformation (Fig. 1) and accordingly, a change in the sensitivity of permeability to confining pressure (Fig. 7b). This mirrors field studies that have shown that aquifers and confining units may develop permeability anisotropy due to the accumulation of inelastic strain following an earthquake 75,76 , a process that can also affect hydrothermal and geothermal systems [77][78][79][80] . Additionally, changes in water level documented in wells after earthquakes have been attributed to changes in permeability 81,82 .
Earthquakes have long been inferred to interact with volcanoes, potentially affecting volcanic activity 83,84 (see 80 for a review). Whilst mechanical oscillations have been suggested to interact with magma 85,86 , here we explore how they could indirectly affect volcanic activity via damage accumulated in the wallrock. At Villarrica volcano in Chile, a moment magnitude (M w ) 8.3 tectonic earthquake rapidly and locally prompted rupture extending to the Earth's surface, and this has been inferred to have prompted the onset of an eruptive episode 17 days later 84 . Similar volcanic unrest was also reported in the following months at nearby Copahue and Nevados de Chillán volcanoes 84 . The trigger of the unrest at these three volcanoes was attributed to the remobilisation of fluid in the adjacent hydrothermal systems (i.e., at a scale much wider than the fault). Here, we posit that the development of permeability anisotropy caused by damage accumulation associated with mechanical oscillations during the earthquake remains a plausible explanation for the inferred redistribution of fluids in the hydrothermal system. Similarly, the M w 9 Tōhoku earthquake in Japan is thought to have modified the stress field around Mount Fuji several hundreds of kilometres away 87 potentially prompting failure of the wall rock around a magmatic storage zone and thus triggering magmatic injection. The accumulation of damage caused by the mechanical oscillations could potentially have enhanced the coalescence of fractures in an already highly stressed (thermal stresses, fluid migration) rock mass.
Earthquakes may also jeopardise the structural stability of prominent rock masses in non-volcanic environments, as demonstrated by observations of widespread landslides and rock-falls that may follow individual earthquake events [88][89][90][91] . It has been suggested that changes in pore fluid pressure [92][93][94] and accumulation of damage 89,95 cause this instability of rock masses, resulting in post-seismic landslides 90 . Here, we advance that the accumulation of inelastic strain and fracture damage during episodes of earthquake activity may, in some instances, amplify creep deformation, prompting sufficient changes to the materials and their pore fluids to increase their susceptibility to undergo wholesale rupture, possibly generating landslides or other types of failure-based hazards. This effect has particularly been observed during similar experiments on variably porous volcanic rocks from Unzen volcano 96 (a volcano in Japan that historically experienced a deadly, large sector collapse event following, in the same day, the 1792 M6.4 Shimabara-Shigatusaku earthquake 97 ): In this study, the authors found that depending on the initial fracture network geometry, mechanical oscillations were not only able to generate damage, but also substantially contribute to the acceleration of the deformation.
Finally, beyond natural environments, mechanical oscillations can affect the stability of anthropogenic structures, such as roads 98 , buildings 99 , bridges 100 , and dams 101 . At their most extreme, large mechanical oscillations, such as those generated by earthquakes, can lead to catastrophic collapse of engineered structures. However, the accumulation of inelastic strain caused by the repetition of low amplitude elastic waves may also severely hamper the resilience of man-made structures. For instance, road traffic can generate highly repetitive, low amplitude elastic waves 102,103 that are able to induce inelastic strain due to damage accumulation in nearby roads, bridges, and buildings; these are important considerations in building regulations as ultimately, mechanical oscillations may weaken constructional materials, and prompt failure 104 .
Despite only using transverse piston motion, our findings indicate that testing the mechanical properties of rocks more frequently under oscillatory stress conditions simulating natural earthquakes would be beneficial in resolving their response and could increase our ability to model the storage of resources in reservoir rocks, as well as the structural stability and collapse of large rock masses, which may generate hazards, such as landslides and volcanic eruptions. In nature, we surmise that this effect may be enhanced by the passage of longitudinal waves creating additional shear in rock masses. We also caution that numerous strategies may be necessary to detect the mechanical weakening of materials caused by oscillatory stresses, and in particular note that increased anisotropy may be a particularly defining attribute of earthquake-damaged materials. Recent studies have shown that rocks with anisotropic fabrics may be particularly susceptible to mechanically induced weakening and  53 , which indicate that repetitive earthquakes that induce anisotropy may perpetuate non-linear, amplifying damage towards wholesale failure, as observed in this study.

Data availability
The datasets generated and/or analysed during the current study are available in the Damage Amplification repository on GitHub, https:// github. com/ Volca nthony/ Damage-ampli ficat ion. www.nature.com/scientificreports/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.